Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.3.6.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $f_0 = f|_{K_0}$ be the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. Then the restriction map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ is a right fibration, and the restriction map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{f_0/}$ is a left fibration.

Proof. Apply Proposition 4.3.6.5 to the inner fibration $q: \operatorname{\mathcal{C}}\rightarrow \Delta ^{0}$. $\square$